Java tutorial
/** * Copyright (C) 2009 - present by OpenGamma Inc. and the OpenGamma group of companies * * Please see distribution for license. */ package com.opengamma.analytics.financial.model.volatility.local; import org.apache.commons.lang.Validate; import org.threeten.bp.ZonedDateTime; import com.opengamma.analytics.financial.model.option.definition.BinomialOptionModelDefinition; import com.opengamma.analytics.financial.model.option.definition.CoxRossRubinsteinBinomialOptionModelDefinition; import com.opengamma.analytics.financial.model.option.definition.EuropeanVanillaOptionDefinition; import com.opengamma.analytics.financial.model.option.definition.OptionDefinition; import com.opengamma.analytics.financial.model.option.definition.StandardOptionDataBundle; import com.opengamma.analytics.financial.model.option.pricing.tree.BinomialOptionModel; import com.opengamma.analytics.financial.model.tree.RecombiningBinomialTree; import com.opengamma.util.time.DateUtils; import com.opengamma.util.time.Expiry; /** * Implementation of the paper by Derman and Kani, The Volatility Smile and its Implied Tree (1994) */ public class DermanKaniImpliedBinomialTreeModel implements ImpliedTreeModel<OptionDefinition, StandardOptionDataBundle> { private static final BinomialOptionModelDefinition<OptionDefinition, StandardOptionDataBundle> CRR = new CoxRossRubinsteinBinomialOptionModelDefinition(); private final int _n; public DermanKaniImpliedBinomialTreeModel(final int n) { Validate.isTrue(n > 0); _n = n; } @Override public ImpliedTreeResult getImpliedTrees(final OptionDefinition definition, final StandardOptionDataBundle data) { Validate.notNull(definition, "definition"); Validate.notNull(data, "data"); final int m1 = RecombiningBinomialTree.NODES.evaluate(_n); final int m2 = RecombiningBinomialTree.NODES.evaluate(_n - 1); final double[][] impliedTree = new double[_n + 1][m1]; //TODO this wastes space final double[] transitionProbabilities = new double[m2]; double[] arrowDebreu = new double[m1]; final double[][] localVolatilityTree = new double[_n][m2]; final double dt = definition.getTimeToExpiry(data.getDate()) / _n; double t = 0; final double spot = data.getSpot(); impliedTree[0][0] = spot; arrowDebreu[0] = 1; int previousNodes = 1; final ZonedDateTime date = data.getDate(); for (int i = 1; i < _n + 1; i++) { final int nodes = RecombiningBinomialTree.NODES.evaluate(i); final BinomialOptionModel<StandardOptionDataBundle> crrModel = new BinomialOptionModel<>(CRR, i); t += dt; final double df1 = Math.exp(dt * data.getInterestRate(t)); final double df2 = Math.exp(dt * data.getCostOfCarry()); final Expiry expiry = new Expiry(DateUtils.getDateOffsetWithYearFraction(date, t)); final int mid = i / 2; if (i % 2 == 0) { impliedTree[i][mid] = spot; addUpperNodes(data, impliedTree, arrowDebreu, i, crrModel, df1, df2, expiry, mid + 1); addLowerNodes(data, impliedTree, arrowDebreu, i, crrModel, df1, df2, expiry, mid - 1); } else { final double c = crrModel .getTreeGeneratingFunction(new EuropeanVanillaOptionDefinition(spot, expiry, true)) .evaluate(data).getNode(0, 0).second; final double sigma = getUpperSigma(impliedTree, arrowDebreu, i - 1, df2, mid + 1); impliedTree[i][mid + 1] = spot * (df1 * c + arrowDebreu[mid] * spot - sigma) / (arrowDebreu[mid] * impliedTree[i - 1][mid] * df2 - df1 * c + sigma); impliedTree[i][mid] = spot * spot / impliedTree[i][mid + 1]; addUpperNodes(data, impliedTree, arrowDebreu, i, crrModel, df1, df2, expiry, mid + 2); addLowerNodes(data, impliedTree, arrowDebreu, i, crrModel, df1, df2, expiry, mid - 1); } for (int j = 0; j < previousNodes; j++) { final double f = impliedTree[i - 1][j] * df2; transitionProbabilities[j] = (f - impliedTree[i][j]) / (impliedTree[i][j + 1] - impliedTree[i][j]); //TODO emcleod 31-8-10 Need to check that transition probabilities are positive - use adjustment suggested in "The Volatility Smile and its Implied Tree" localVolatilityTree[i - 1][j] = Math .sqrt(transitionProbabilities[j] * (1 - transitionProbabilities[j])) * Math.log(impliedTree[i][j + 1] / impliedTree[i][j]); //TODO need 1/sqrt(dt) here } final double[] temp = new double[m1]; temp[0] = (1 - transitionProbabilities[0]) * arrowDebreu[0] / df1; temp[nodes - 1] = (transitionProbabilities[previousNodes - 1] * arrowDebreu[previousNodes - 1]) / df1; for (int j = 1; j < nodes - 1; j++) { temp[j] = (transitionProbabilities[j - 1] * arrowDebreu[j - 1] + (1 - transitionProbabilities[j]) * arrowDebreu[j]) / df1; } arrowDebreu = temp; previousNodes = nodes; } final Double[][] impliedTreeResult = new Double[_n + 1][m1]; final Double[][] localVolResult = new Double[_n][m2]; for (int i = 0; i < impliedTree.length; i++) { for (int j = 0; j < impliedTree[i].length; j++) { impliedTreeResult[i][j] = impliedTree[i][j]; if (i < _n && j < m2) { localVolResult[i][j] = localVolatilityTree[i][j]; } } } return new ImpliedTreeResult(new RecombiningBinomialTree<>(impliedTreeResult), new RecombiningBinomialTree<>(localVolResult)); } private void addLowerNodes(final StandardOptionDataBundle data, final double[][] impliedTree, final double[] arrowDebreu, final int step, final BinomialOptionModel<StandardOptionDataBundle> crrModel, final double df1, final double df2, final Expiry expiry, final int mid) { double sigma = getLowerSigma(impliedTree, arrowDebreu, step - 1, df2, mid); for (int i = mid; i >= 0; i--) { final double p = crrModel .getTreeGeneratingFunction( new EuropeanVanillaOptionDefinition(impliedTree[step - 1][i], expiry, false)) .evaluate(data).getNode(0, 0).second; final double forward = impliedTree[step - 1][i] * df2; impliedTree[step][i] = (impliedTree[step][i + 1] * (df1 * p - sigma) + arrowDebreu[i] * impliedTree[step - 1][i] * (forward - impliedTree[step][i + 1])) / (df1 * p - sigma + arrowDebreu[i] * (forward - impliedTree[step][i + 1])); if (i > 0) { sigma -= arrowDebreu[i - 1] * (impliedTree[step - 1][i] - impliedTree[step - 1][i - 1] * df2); } } } private void addUpperNodes(final StandardOptionDataBundle data, final double[][] impliedTree, final double[] arrowDebreu, final int step, final BinomialOptionModel<StandardOptionDataBundle> crrModel, final double df1, final double df2, final Expiry expiry, final int mid) { double sigma = getUpperSigma(impliedTree, arrowDebreu, step - 1, df2, mid); for (int i = mid; i < RecombiningBinomialTree.NODES.evaluate(step); i++) { final double c = crrModel .getTreeGeneratingFunction( new EuropeanVanillaOptionDefinition(impliedTree[step - 1][i - 1], expiry, true)) .evaluate(data).getNode(0, 0).second; final double forward = impliedTree[step - 1][i - 1] * df2; impliedTree[step][i] = (impliedTree[step][i - 1] * (df1 * c - sigma) - arrowDebreu[i - 1] * impliedTree[step - 1][i - 1] * (forward - impliedTree[step][i - 1])) / (df1 * c - sigma - arrowDebreu[i - 1] * (forward - impliedTree[step][i - 1])); sigma -= arrowDebreu[i] * (impliedTree[step - 1][i] * df2 - impliedTree[step - 1][i - 1]); } } private double getLowerSigma(final double[][] impliedTree, final double[] arrowDebreu, final int previousStep, final double df2, final int start) { double sigma = 0; for (int i = start - 1; i >= 0; i--) { sigma += arrowDebreu[i] * (impliedTree[previousStep][start] - impliedTree[previousStep][i] * df2); } return sigma; } private double getUpperSigma(final double[][] impliedTree, final double[] arrowDebreu, final int previousStep, final double df2, final int start) { double sigma = 0; for (int i = start; i < RecombiningBinomialTree.NODES.evaluate(previousStep + 1); i++) { sigma += arrowDebreu[i] * (impliedTree[previousStep][i] * df2 - impliedTree[previousStep][start - 1]); } return sigma; } }